I am looking for a reference where it is proven that the exponential map described above is surjective. Here, I am denoting by $\mathfrak{so}(1,n)$ the Lie Algebra of the group $SO(1,n)$.
So we have that \begin{align} SO(1,n) & = \Big\{A\in \mathbb{R}^{n\times 1, n\times 1}:AHA^t=H, \det A = 1\Big\} \\ \mathfrak{so}(1,n) & = \Big\{ B\in\mathbb{R}^{n\times1, n\times1}: BH + HB^t = 0\Big\} \end{align} And also $H = diag(1,-1,-1,...,-1)$ And $\exp$ is the matrix exponential. Now, I know this is true but I'm specifically looking for a book or paper where this is proven.
I am not an expert on Lie groups but here is what I recently learned. Let me denote $G = O(n,1)$.
First of all, the image of $\exp$ is at best $G_0$, the identity component of $G$, which is not $SO(n,1)$. $G$ has four connected components, distinguished according to whether the corresponding isometries of $\mathbb{R}^{n,1}$ preserve the orientation of space and time. The identity component is $G_0 = O^{++}(n,1)$ (also denoted $SO^+(n,1)$). It is an index 2 subgroup of $SO(n,1)$.
That being said, it is true that $\exp$ is onto $G_0$. This is however not an easy result. Its "history" is discussed in the book of Gallier and Quaintance (GQ) Differential geometry and Lie groups—a computational perspective. They explain that the result has been proved by Nishikawa (1983), based on results of Burgoyne and Cushman (1977) and Djoković (1980); and independently by Riesz in 1957 (published in 1993). GQ comment that both proofs add up to several dozen pages. They proceed to prove the special cases $n=3$ and $n=2$, which are easier thanks to the exceptional isomorphisms $O^{++}(3,1) \approx PSL(2, \mathbb{C})$ and $O^{++}(2,1) \approx PSL(2, \mathbb{R})$.
I am writing a book on hyperbolic geometry where I plan to discuss this (you can check out a preliminary version of my book here). It seems to me that the result can also be derived from the classification of isometries (recall that $O^{++}(n,1) \approx \operatorname{Isom}^+(\mathbb{H}^n)$): it should not be too hard to check that any elliptic, parabolic, or hyperbolic isometry belongs to a one-parameter subgroup. Obtaining the classification is not trivial though: it relies on either the Jordan--Chevalley (or the KAN) decomposition of $O(n,1)$, for which there is no simple proof as far as I can tell (meaning avoiding algebraic geometry), or knowing the classification of isometries à la Gromov, which is not too hard but requires a lot of background. If anyone has more insights to share, I'd be delighted to hear them.
Out of interest: Nishikawa also proves that the exponential map is not surjective for $G = O_0(p,q)$ as soon as $p, q\geqslant 2$. However it is known since Sibuya (1960) that the square of any element of $O(p,q)$ is in the image of $\exp$.